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An Optimization-Driven Approach for Modeling AS-level Internet Connectivity
Presented by:Hyunseok Chang
Joint work with Sugih Jamin (UM) and Walter Willinger (AT&T)
IPAM 2003
AS-level Internet graph
Autonomous System (AS)Peering relationship
Provider-customer typePeer-to-peer type
Autonomous System (AS) Point of Presence (PoP)
IPAM 2003
Inferred Internet AS graph
Highly variable AS vertex degree distribution.
IPAM 2003
Related research Generating Internet-like random graphs
Focusing on the quality of a generated graph, not on the generation process itself.
e.g., Inet generator. Modeling the Internet AS graph
A graph generation process reflects actual Internet growth history.
e.g., Barabasi-Albert model, Fabrikant-HOT model.
Our study focuses on the modeling aspect.
IPAM 2003
Our research focus The related works have been very generic, based
only on topological properties (e.g., node degree). Our starting point: Fabrikant-HOT (Heuristically Op
timized Trade-off) model for Internet growth.
Attempt to explain how inter-AS peering relationships are established in an optimization-driven fashion.
IPAM 2003
AS degree distributions
At first, we focus on PC subgraph single-homed.
IPAM 2003
Fabrikant-HOT model Each new node solves the local optimization p
roblem to find a target node to connect to. Each new node i connects to an existing node
j that minimizes the weighted sum of two objectives: min (dij + hj) dij (last mile cost) = Euclidean distance from i to j hj (transmission delay cost) = average hop distanc
e from j to all other nodes
IPAM 2003
Multi-PoP ISPs Fabrikant-HOT model assumes each node has
a single PoP, whereas ISPs maintain multiple PoPs.
In reality, dij and hj may not be independent.
AS# Name # of PoPs
2914 Verio 121
7018 AT&T 108
1221 Telstra 61
3356 Level3 52
1239 SprintLink 43
IPAM 2003
Modified Fabrikant-HOT model Each node maintains multiple PoPs. The number of PoPs of an existing node increa
ses over time.
IPAM 2003
Original Fabrikant-HOT model
ij jd h
For each new node i,
Find node j that minimizes Connect node i to node j.
( )l loc set j jild hmin
For each existing node u, increment |loc-set(u)| with prob. Krank(u)-.
|loc-set(i)| 1
Modified
IPAM 2003
Modified Fabrikant-HOT model creates a hot spot!
(a) Fabrikant-HOT model (b) Modified Fabrikant-HOT model
IPAM 2003
Our proposed models Univariate HOT model.
Criteria: (i) AS geography. Bivariate HOT model.
Criteria: (i) AS geography, (ii) AS business model. Various extensions.
IPAM 2003
Our proposed models Univariate HOT model
Criteria: (i) AS geography. Bivariate HOT model
Criteria: (i) AS geography, (ii) AS business model. Various extensions.
IPAM 2003
Univariate HOT model A single-objective optimization: minimize last-
mile connection cost. A newly arriving node i connects to an existing
node that has the closest PoP to i. An existing node u gradually increases the nu
mber of PoPs as later arriving nodes are attached to u.
IPAM 2003
Modified Fabrikant-HOT model
For each new node i, |loc-set(i)| 1 Find node j that minimizes Connect node i to node j. For each existing node u, increment |loc-set(u)| with prob. Krank(u)-.
( )l loc set j jild hmin
Univariate HOT model
IPAM 2003
Node size & degree distribution =0.1
Exponential-type distribution for the number of locations per node.
IPAM 2003
Node size & degree distribution =1.0
Highly-variable distribution for the number of locations per node.
IPAM 2003
AS size vs. AS degree Univariate model predicts that AS degree varia
bility would be comparable to AS size (i.e., # of PoPs) variability.
However, in the current Internet: Maximum AS degree ~ 103
Maximum # of PoPs per AS ~ 102
Q: Are there other criteria?
IPAM 2003
Proposed HOT models Univariate HOT model.
Criteria: (i) AS geography. Bivariate HOT model.
Criteria: (i) AS geography, (ii) AS business model. Various extensions.
IPAM 2003
Bivariate HOT model What if new AS i has multiple candidate provid
ers in close geographic proximity? e.g., global criteria set = {reliability, cost, cu
stomer service} and customer i’s local criteria set (i) = {reliability, cost}, ISP X: {99%, $100/Mbps, fair} ISP Y: {98%, $150/Mbps, good} ISP Z: {97%, $50/Mbps, bad}With respect to (i), ISP X and Z are Pareto optimal.
X >(i) Y
Y < >(i) Z
X < >(i) Z
IPAM 2003
Bivariate HOT model Given a new node i, Initialize nbr-set(i) as containing all the nodes
which have a PoP in close proximity to i. Remove any node in nbr-set(i), which is not P
areto-optimal in terms of (i); an existing node u has quality vector Q(u)=(x1,…,xN).
Connect node i to one randomly selected node from nbr-set(i).
IPAM 2003
Node size & degree distribution
Bivariate model matches the Internet well!
IPAM 2003
Proposed HOT models Univariate HOT model.
Criteria: (i) AS geography. Bivariate HOT model.
Criteria: (i) AS geography, (ii) AS business model. Various extensions.
IPAM 2003
Extension #1: multiple providers per AS Observation: # of providers for an AS increase
s over time (similar to # of PoPs).
In our original model: Every time a new node i is added to a graph,
each existing node u gets a chance to: i) increment |loc-set(u)| with prob. Krank(u)-, ii) increment |prov-set(u)|with prob. Rrank(u)- (R<<K).
In our extended model:
IPAM 2003
Extension #2:peer-to-peer neighbors Observation: decision on providers is unilatera
l, but decision on peers is bilateral. For existing nodes u and v to become peering
partners, we expect: i) unbr-set(v) (or, vnbr-set(u)), ii) u (u) v and v (v) u (u), (v) : peering criteria set for u and v.
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Extension #3: AS evolution Node death & change-of-provider
events. Role transition (e.g., provider peer). Evolving qualities.
IPAM 2003
Summary Our modeling approach:
Explores the possibility of studying the Internet AS evolution in an optimization-based framework.
Introduces domain-specific concepts in the modeling framework.
Challenges: Model validation Application(s)
Any kind of input is welcome!!